Explanation Procedure used: The general solution of the homogeneous differential equation a y ″ + b y ′ + c y = G ( x ) can be written as y ( x ) = y c ( x ) + y p ( x ) , where y c ( x ) is complimentary solution of a y ″ + b y ′ + c y = 0 and y p ( x ) is the particular solution of a y ″ + b y ′ + c y = G ( x ) . The complimentary solution is y c = C 1 y 1 ( x ) + C 2 y 2 ( x ) . (1) Here C 1 and C 2 are constants. The parameters C 1 and C 2 in y c can be replaced with functions u 1 and u 2 (variable parameters), so that y p ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) (2) Determine two unknown functions u 1 and u 2 , by Cramer’s Rule, y 1 u ′ 1 + y 2 u ′ 2 = 0 (3) y 1 u ′ 1 + y 2 u ′ 2 = f ( x ) (4) Use Wronskian method to find W , W 1 and W 2 . W = | y 1 y 2 y ′ 1 y ′ 2 | (5) W 1 = | 0 y 2 f ( x ) y ′ 2 | W 2 = | y 1 0 y ′ 1 f ( x ) | u ′ 1 = W 1 W = − y 2 f ( x ) W and u ′ 2 = W 2 W = y 1 f ( x ) W (6) The functions u 1 and u 2 can be found by integrating the results in (6). If the integrand is continuous on the interval [ x 0 , x ] , u 1 ( x ) = ∫ x 0 x y 2 ( t ) f ( t ) W ( t ) d t and u 2 ( x ) = ∫ x 0 x y 1 ( t ) f ( t ) W ( t ) d t (7) Substitute the values of u 1 and u 2 in equation (2) to find the particular solution. The general solution of the equation is then y = y c + y p . Calculations: The auxiliary equation for the associated homogeneous equation y ″ + y ′ − 2 y = ln x is m 2 + m − 2 = 0 . The roots are m 1 = 1 and m 2 = − 2

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.6, Problem 25E

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To solve: The given differential equation y″+y′−2y=lnx by variation of parameters.

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Chapter 4.6, Problem 24E

Chapter 4.6, Problem 26E

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Given that x(t) = c1 cos t + c2 sin t is the...Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The given differential equation y ″ + y ′ − 2 y = ln x by variation of parameters.

Solution Summary: The author explains the general solution of the homogeneous differential equation ay′′+b y -2y=mathrmlnx by variation of parameters.

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To solve: The given differential equation y″+y′−2y=lnx by variation of parameters.

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